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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Arcs defined by one-parameter semigroups of operators


Authors: Hugo D. Junghenn and C. T. Taam
Journal: Proc. Amer. Math. Soc. 44 (1974), 113-120
MSC: Primary 47D05
MathSciNet review: 0331114
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Abstract: Let $ T(t)(t \geqq 0)$ be a one-parameter semigroup of continuous linear operators in a locally convex reflexive linear topological space $ X$ such that $ T(c)$ is an isomorphism (into) for some $ c > 0$. It is proved that for any $ x \in X,T( \cdot )x$ is of bounded variation on finite intervals if and only if $ x$ is in the domain of the infinitesimal generator of $ T(t)$. The result is interpreted geometrically in terms of arc-length.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0331114-7
Keywords: Semigroup of operators, reflexive, arc-length, bounded variation, absolute continuity, local equicontinuity
Article copyright: © Copyright 1974 American Mathematical Society